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<h1 class="mume-header" id="glove-and-word-senses">GloVe and Word Senses</h1>

<ul>
<li><a href="#glove-and-word-senses">GloVe and Word Senses</a>
<ul>
<li><a href="#1-glove-global-vectors-for-word-representation">1. GloVe: Global Vectors for Word Representation</a>
<ul>
<li><a href="#1-%E4%B9%8B%E5%89%8D%E7%9A%84%E8%AF%8D%E5%90%91%E9%87%8F%E6%96%B9%E6%B3%95">1. &#x4E4B;&#x524D;&#x7684;&#x8BCD;&#x5411;&#x91CF;&#x65B9;&#x6CD5;</a></li>
<li><a href="#2-glove%E6%A8%A1%E5%9E%8B">2. GloVe&#x6A21;&#x578B;</a></li>
<li><a href="#3-%E6%A8%A1%E5%9E%8B%E5%85%B3%E7%B3%BB">3. &#x6A21;&#x578B;&#x5173;&#x7CFB;</a></li>
</ul>
</li>
</ul>
</li>
</ul>
<hr>
<h2 class="mume-header" id="1-glove-global-vectors-for-word-representation">1. GloVe: Global Vectors for Word Representation</h2>

<h3 class="mume-header" id="1-%E4%B9%8B%E5%89%8D%E7%9A%84%E8%AF%8D%E5%90%91%E9%87%8F%E6%96%B9%E6%B3%95">1. &#x4E4B;&#x524D;&#x7684;&#x8BCD;&#x5411;&#x91CF;&#x65B9;&#x6CD5;</h3>

<ol>
<li>Matrix Factorization Methods<br>
global matrix factorization(&#x5168;&#x5C40;&#x7684;&#x77E9;&#x9635;&#x5206;&#x89E3;)<br>
&#x57FA;&#x4E8E;&#x5171;&#x73B0;&#x77E9;&#x9635;&#x7684;&#x65B9;&#x6CD5;&#xFF1A;&#x9996;&#x5148;&#x83B7;&#x53D6;&#x6240;&#x6709;&#x5355;&#x8BCD;&#x95F4;&#x7684;&#x5171;&#x73B0;&#x77E9;&#x9635;(co-occurrence)&#xFF0C;&#x7136;&#x540E;&#x4F7F;&#x7528;&#x5947;&#x5F02;&#x503C;&#x5206;&#x89E3;(SVD)&#x8FDB;&#x884C;&#x964D;&#x7EF4;&#x5F97;&#x5230;&#x8BCD;&#x5411;&#x91CF;</li>
</ol>
<blockquote>
<p>&#x4F18;&#x70B9;&#xFF1A;&#x53EF;&#x4EE5;&#x5229;&#x7528;&#x5168;&#x5C40;&#x7684;&#x7EDF;&#x8BA1;&#x4FE1;&#x606F;(golbal statistical information)<br>
&#x7F3A;&#x70B9;&#xFF1A;&#x5728;&#x5355;&#x8BCD;&#x7C7B;&#x522B;&#x4EFB;&#x52A1;(word analogy)&#x4E0A;&#x8868;&#x73B0;&#x5F88;&#x5DEE;&#xFF0C;&#x5373;&#x65E0;&#x6CD5;&#x6709;&#x6548;&#x5730;&#x6355;&#x6349;&#x5355;&#x8BCD;&#x95F4;&#x7684;&#x8BED;&#x4E49;&#x5173;&#x7CFB;</p>
</blockquote>
<p>&#x57FA;&#x4E8E;&#x77E9;&#x9635;&#x5206;&#x89E3;&#x7684;&#x65B9;&#x6CD5;&#x90FD;&#x662F;&#x4F7F;&#x7528;&#x4E86;<strong>&#x4F4E;&#x79E9;&#x8FD1;&#x4F3C;low-rank approximations</strong>&#x7684;&#x539F;&#x7406;&#x6765;&#x5206;&#x89E3;&#x5927;&#x7684;&#x77E9;&#x9635;&#x6765;&#x83B7;&#x53D6;&#x8BED;&#x6599;&#x4E2D;&#x7684;&#x7EDF;&#x8BA1;&#x4FE1;&#x606F;&#xFF0C;&#x4E0D;&#x8FC7;&#x4E0D;&#x540C;&#x7684;&#x65B9;&#x6CD5;&#x4E2D;&#x77E9;&#x9635;&#x7684;&#x5F62;&#x5F0F;&#x662F;&#x4E0D;&#x540C;&#x7684;&#xFF0C;&#x5982;&#x5728;latent semantic analysis (LSA)&#x4E2D;&#xFF0C;&#x77E9;&#x9635;&#x662F;&quot;term-doucment&quot;&#x7684;&#x5F62;&#x5F0F;&#x3002;</p>
<ol start="2">
<li>Shallow Window-Based Methods<br>
local context window(&#x5C40;&#x90E8;&#x7A97;&#x53E3;)&#x6765;&#x9884;&#x6D4B;&#x5355;&#x8BCD;<br>
Mikolov&#x8BBA;&#x6587;&#x4E2D;&#x63D0;&#x51FA;&#x7684;skip-gram&#x548C;continuous bag-of-words&#x5229;&#x7528;&#x4E0A;&#x4E0B;&#x6587;&#x548C;&#x4E2D;&#x5FC3;&#x8BCD;&#x5173;&#x7CFB;&#x6765;&#x5B66;&#x4E60;&#x8BCD;&#x5411;&#x91CF;&#xFF0C;&#x8FD9;&#x4E9B;&#x5411;&#x91CF;&#x53EF;&#x4EE5;&#x5B66;&#x4E60;&#x5230;&#x8BED;&#x8A00;&#x6A21;&#x5F0F;(linguistic patterns)&#x548C;&#x5355;&#x8BCD;&#x5411;&#x91CF;&#x95F4;&#x7684;&#x7EBF;&#x6027;&#x5173;&#x7CFB;(linear relationships between word vectors)&#xFF0C;&#x8FD9;&#x91CC;&#x8BED;&#x8A00;&#x6A21;&#x5F0F;&#x53EF;&#x4EE5;&#x5305;&#x542B;&#x53E5;&#x6CD5;&#x548C;&#x8BED;&#x4E49;(syntasitc and semantic)&#x3002;</li>
</ol>
<blockquote>
<p>&#x4F18;&#x70B9;&#xFF1A;&#x5728;&#x5355;&#x8BCD;&#x7C7B;&#x522B;&#x4EFB;&#x52A1;&#x4E2D;&#x8868;&#x73B0;&#x826F;&#x597D;&#xFF0C;&#x5373;&#x53EF;&#x4EE5;&#x6709;&#x6548;&#x5730;&#x6355;&#x6349;&#x5355;&#x8BCD;&#x7684;&#x8BED;&#x4E49;&#xFF0C;&#x6362;&#x8A00;&#x4E4B;&#x5C31;&#x662F;&#x5728;&#x5355;&#x8BCD;&#x7A7A;&#x95F4;&#x4E2D;&#x53EF;&#x4EE5;&#x5305;&#x542B;&#x6709;&#x8BED;&#x4E49;&#x4FE1;&#x606F;&#x7684;&#x5B50;&#x7ED3;&#x6784;<br>
&#x7F3A;&#x70B9;&#xFF1A;&#x6A21;&#x578B;&#x662F;&#x901A;&#x8FC7;&#x904D;&#x5386;&#x8BED;&#x6599;&#x7684;&#x5F62;&#x5F0F;&#x6765;&#x5B66;&#x4E60;&#x8BCD;&#x5411;&#x91CF;&#xFF0C;&#x65E0;&#x6CD5;&#x5229;&#x7528;&#x5168;&#x5C40;&#x7684;&#x8BED;&#x6599;&#x7EDF;&#x8BA1;&#x4FE1;&#x606F;</p>
</blockquote>
<h3 class="mume-header" id="2-glove%E6%A8%A1%E5%9E%8B">2. GloVe&#x6A21;&#x578B;</h3>

<p>&#x4E4B;&#x524D;&#x7684;&#x4E24;&#x79CD;&#x65B9;&#x6CD5;&#x5404;&#x6709;&#x5229;&#x5F0A;&#xFF0C;&#x6240;&#x4EE5;GloVe&#x6A21;&#x578B;&#x662F;&#x7ED3;&#x5408;&#x5176;&#x5404;&#x81EA;&#x7684;&#x4F18;&#x70B9;&#x5F97;&#x51FA;&#x7684;<br>
&#x601D;&#x8DEF;&#xFF1A;</p>
<ul>
<li>&#x901A;&#x8FC7;&#x76F4;&#x63A5;&#x5728;<strong>&#x5171;&#x73B0;&#x77E9;&#x9635;</strong>&#x4E0A;&#x8BA1;&#x7B97;&#x6765;&#x5229;&#x7528;&#x8BED;&#x6599;&#x7684;&#x5168;&#x5C40;&#x7EDF;&#x8BA1;&#x4FE1;&#x606F;</li>
<li>&#x76F4;&#x63A5;&#x8BA1;&#x7B97;<strong>&#x4E0A;&#x4E0B;&#x6587;&#x7684;&#x76F8;&#x5173;&#x6982;&#x7387;</strong>&#x6765;&#x5B9E;&#x73B0;&#x5229;&#x7528;&#x5C40;&#x90E8;&#x4E0A;&#x4E0B;&#x6587;&#x4FE1;&#x606F;</li>
</ul>
<p>global log-bilinear regression(&#x5168;&#x5C40;&#x5BF9;&#x6570;&#x53CC;&#x7EBF;&#x6027;&#x56DE;&#x5F52;)</p>
<p>&#x7B26;&#x53F7;&#x58F0;&#x660E;&#xFF1A;</p>
<blockquote>
<p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.07847em;">X</span></span></span></span>&#x8868;&#x793A;&#x5355;&#x8BCD;&#x7684;&#x5171;&#x73B0;&#x77E9;&#x9635;&#x6216;&#x8BA1;&#x6570;(word-word co-occurrence counts)<br>
<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>X</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">X_{ij}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span>&#x4EE3;&#x8868;&#x5355;&#x8BCD;<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.05724em;">j</span></span></span></span>&#x51FA;&#x73B0;&#x5728;&#x5355;&#x8BCD;<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord mathdefault">i</span></span></span></span>&#x4E0A;&#x4E0B;&#x6587;&#x4E2D;&#x7684;&#x6B21;&#x6570;<br>
<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>X</mi><mi>i</mi></msub><mo>=</mo><msub><mo>&#x2211;</mo><mi>k</mi></msub><msub><mi>X</mi><mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow><annotation encoding="application/x-tex">X_i = \sum_k X_{ik}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.0497100000000001em;vertical-align:-0.29971000000000003em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em;">&#x2211;</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1863979999999999em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.29971000000000003em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>&#x4EE3;&#x8868;&#x4EFB;&#x610F;&#x5355;&#x8BCD;<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.03148em;">k</span></span></span></span>&#x51FA;&#x73B0;&#x5728;&#x5355;&#x8BCD;<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord mathdefault">i</span></span></span></span>&#x4E0A;&#x4E0B;&#x6587;&#x4E2D;&#x7684;&#x6B21;&#x6570;&#x603B;&#x548C;<br>
<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>P</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mi>P</mi><mo>(</mo><mi>j</mi><mo>&#x2223;</mo><mi>i</mi><mo>)</mo><mo>=</mo><mfrac><msub><mi>X</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><msub><mi>X</mi><mi>i</mi></msub></mfrac></mrow><annotation encoding="application/x-tex">P_{ij}=P(j \mid i)=\frac{X_{ij}}{X_i}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.05724em;">j</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&#x2223;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">i</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.4307509999999999em;vertical-align:-0.44509999999999994em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.985651em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3280857142857143em;"><span style="top:-2.357em;margin-left:-0.07847em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.50732em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3280857142857143em;"><span style="top:-2.357em;margin-left:-0.07847em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.2818857142857143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.44509999999999994em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>&#x662F;&#x5355;&#x8BCD;<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.05724em;">j</span></span></span></span>&#x51FA;&#x73B0;&#x5728;&#x5355;&#x8BCD;<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord mathdefault">i</span></span></span></span>&#x4E0A;&#x4E0B;&#x6587;&#x4E2D;&#x7684;&#x6982;&#x7387;</p>
</blockquote>
<p><strong>&#x6B65;&#x9AA4;&#xFF1A;</strong><br>
(1) &#x4F7F;&#x7528;&#x5171;&#x73B0;&#x6982;&#x7387;&#x7684;&#x6BD4;&#x7387;&#x6765;&#x63CF;&#x8FF0;&#x8BE5;&#x6A21;&#x578B;<br>
Glove&#x6A21;&#x578B;&#x8BA4;&#x4E3A;&#x4F7F;&#x7528;&#x5171;&#x73B0;&#x6982;&#x7387;&#x7684;&#x6BD4;&#x503C;&#x6BD4;&#x76F4;&#x63A5;&#x4F7F;&#x7528;&#x5171;&#x73B0;&#x6982;&#x7387;&#x7684;&#x6548;&#x679C;&#x597D;&#xFF0C;&#x5177;&#x4F53;&#x770B;&#x539F;&#x6587;&#x4E2D;ice&#x548C;steam&#x6982;&#x7387;&#x548C;&#x6BD4;&#x7387;&#x7684;&#x8868;&#x683C;1&#x5982;&#x4E0B;&#xFF1A;<br>
<img src="imgs/ratio_co_coourrence.jpg" alt="ratio"><br>
&#x4F7F;&#x7528;&#x4E0B;&#x9762;&#x7684;&#x516C;&#x5F0F;&#x6765;&#x8868;&#x793A;&#x5171;&#x73B0;&#x6982;&#x7387;&#x7684;&#x6BD4;&#x503C;&#xFF1A;<br>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mtable side="right"><mlabeledtr><mtd><mtext>(1)</mtext></mtd><mtd><mrow><mi>F</mi><mrow><mo fence="true">(</mo><msub><mi>w</mi><mi>i</mi></msub><mo separator="true">,</mo><msub><mi>w</mi><mi>j</mi></msub><mo separator="true">,</mo><msub><mover accent="true"><mi>w</mi><mo>~</mo></mover><mi>k</mi></msub><mo fence="true">)</mo></mrow><mo>=</mo><mfrac><msub><mi>P</mi><mrow><mi>i</mi><mi>k</mi></mrow></msub><msub><mi>P</mi><mrow><mi>j</mi><mi>k</mi></mrow></msub></mfrac></mrow></mtd></mlabeledtr></mtable><annotation encoding="application/x-tex">F\left(w_{i}, w_{j}, \tilde{w}_{k}\right)=\frac{P_{i k}}{P_{j k}} \tag{1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.036108em;vertical-align:-0.286108em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">F</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span></span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.16666em;">~</span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.332438em;vertical-align:-0.972108em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.36033em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.972108em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="tag"><span class="strut" style="height:2.332438em;vertical-align:-0.972108em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">1</span></span><span class="mord">)</span></span></span></span></span></span><br>
&#x5176;&#x4E2D;<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>w</mi><mo>&#x2208;</mo><msup><mi>R</mi><mi>d</mi></msup></mrow><annotation encoding="application/x-tex">w \in R^d</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&#x2208;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.849108em;vertical-align:0em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.849108em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span></span></span></span></span></span></span></span>&#x662F;&#x8BCD;&#x5411;&#x91CF;&#xFF0C;<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mover accent="true"><mi>w</mi><mo>~</mo></mover><mo>&#x2208;</mo><msup><mi>R</mi><mi>d</mi></msup></mrow><annotation encoding="application/x-tex">\tilde{w} \in R^d</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7069599999999999em;vertical-align:-0.0391em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span></span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.16666em;">~</span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&#x2208;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.849108em;vertical-align:0em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.849108em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span></span></span></span></span></span></span></span>&#x662F;<strong>&#x5206;&#x5F00;&#x7684;&#x4E0A;&#x4E0B;&#x6587;&#x8BCD;&#x5411;&#x91CF;</strong>&#xFF0C;&#x5373;&#x4E0A;&#x8FF0;3&#x4E2A;&#x8BCD;&#x5411;&#x91CF;&#x6765;&#x81EA;&#x4E8E;&#x4E24;&#x4E2A;&#x8BCD;&#x5D4C;&#x5165;&#x77E9;&#x9635;&#xFF0C;&#x7B49;&#x53F7;&#x53F3;&#x8FB9;&#x662F;&#x76F4;&#x63A5;&#x66F4;&#x52A0;&#x5171;&#x73B0;&#x77E9;&#x9635;&#x6C42;&#x5F97;&#x7684;&#xFF0C;<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>P</mi><mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow><annotation encoding="application/x-tex">P_{i k}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>&#x548C;<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>P</mi><mrow><mi>j</mi><mi>k</mi></mrow></msub></mrow><annotation encoding="application/x-tex">P_{j k}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span>&#x5206;&#x522B;&#x662F;&#x5728;&#x4E2D;&#x5FC3;&#x8BCD;&#x4E3A;<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord mathdefault">i</span></span></span></span>&#x548C;<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.05724em;">j</span></span></span></span>&#x65F6;&#x4E0A;&#x4E0B;&#x6587;&#x662F;<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.03148em;">k</span></span></span></span>&#x7684;&#x6982;&#x7387;&#x3002;<br>
(2)&#x786E;&#x5B9A;&#x51FD;&#x6570;<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">F</span></span></span></span>&#x7684;&#x5F62;&#x5F0F;<br>
<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">F</span></span></span></span>&#x51FD;&#x6570;&#x4E3A;&#x4E86;&#x4E3A;&#x4E86;&#x5728;&#x8BCD;&#x5411;&#x91CF;&#x7A7A;&#x95F4;&#x8868;&#x793A;&#x51FA;&#x6BD4;&#x7387;<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>P</mi><mrow><mi>i</mi><mi>k</mi></mrow></msub><mi mathvariant="normal">/</mi><msub><mi>P</mi><mrow><mi>j</mi><mi>k</mi></mrow></msub></mrow><annotation encoding="application/x-tex">P_{i k}/P_{j k}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.036108em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">/</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span>&#xFF0C;<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>w</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">w_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>&#x548C;<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>w</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">w_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span>&#x505A;&#x5DEE;&#xFF0C;&#x7136;&#x540E;&#x4E0E;<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mover accent="true"><mi>w</mi><mo>~</mo></mover></mrow><annotation encoding="application/x-tex">\tilde{w}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6678599999999999em;vertical-align:0em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span></span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.16666em;">~</span></span></span></span></span></span></span></span></span>&#x505A;&#x5185;&#x79EF;&#xFF0C;&#x8FD9;&#x79CD;&#x65B9;&#x5F0F;&#x53EF;&#x4EE5;&#x66F4;&#x597D;&#x4F53;&#x73B0;&#x51FD;&#x6570;<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">F</span></span></span></span>&#x7684;&#x7EBF;&#x6027;&#xFF0C;&#x5E76;&#x4E14;&#x66F4;&#x597D;&#x5C06;&#x8BCD;&#x5411;&#x91CF;&#x5404;&#x4E2A;&#x7EF4;&#x5EA6;&#x5BF9;&#x5E94;&#x8D77;&#x6765;&#xFF0C;&#x516C;&#x5F0F;&#x5982;&#x4E0B;&#xFF1A;<br>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mtable side="right"><mlabeledtr><mtd><mtext>(2)</mtext></mtd><mtd><mrow><mi>F</mi><mrow><mo fence="true">(</mo><msup><mrow><mo fence="true">(</mo><msub><mi>w</mi><mi>i</mi></msub><mo>&#x2212;</mo><msub><mi>w</mi><mi>j</mi></msub><mo fence="true">)</mo></mrow><mi>T</mi></msup><msub><mover accent="true"><mi>w</mi><mo>~</mo></mover><mi>k</mi></msub><mo fence="true">)</mo></mrow><mo>=</mo><mfrac><msub><mi>P</mi><mrow><mi>i</mi><mi>k</mi></mrow></msub><msub><mi>P</mi><mrow><mi>j</mi><mi>k</mi></mrow></msub></mfrac></mrow></mtd></mlabeledtr></mtable><annotation encoding="application/x-tex">F\left(\left(w_{i}-w_{j}\right)^{T} \tilde{w}_{k}\right)=\frac{P_{i k}}{P_{j k}} \tag{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.80002em;vertical-align:-0.65002em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">F</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size2">(</span></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">&#x2212;</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.981231em;"><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span></span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.16666em;">~</span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size2">)</span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.332438em;vertical-align:-0.972108em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.36033em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.972108em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="tag"><span class="strut" style="height:2.332438em;vertical-align:-0.972108em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">2</span></span><span class="mord">)</span></span></span></span></span></span><br>
&#x5728;&#x8BED;&#x6599;&#x4E2D;&#x6BCF;&#x4E00;&#x4E2A;&#x8BCD;&#x5373;&#x53EF;&#x4EE5;&#x662F;&#x4E2D;&#x5FC3;&#x8BCD;&#x4E5F;&#x53EF;&#x4EE5;&#x662F;&#x4E0A;&#x4E0B;&#x6587;&#xFF0C;&#x6240;&#x4EE5;&#x5355;&#x8BCD;&#x5171;&#x73B0;&#x77E9;&#x9635;<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.07847em;">X</span></span></span></span>&#x662F;&#x5BF9;&#x79F0;&#x77E9;&#x9635;&#x7684;&#xFF0C;&#x56E0;&#x6B64;&#x6A21;&#x578B;&#x9700;&#x8981;&#x4FDD;&#x8BC1;&#x5404;&#x4E2A;&#x8BCD;&#x5411;&#x91CF;&#x662F;&#x53EF;&#x4EA4;&#x6362;&#x7684;&#xFF0C;&#x51FD;&#x6570;<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">F</span></span></span></span>&#x662F;&#x540C;&#x6001;&#x7684;(<a href="https://zh.wikipedia.org/wiki/%E5%90%8C%E6%80%81">homomorphism</a>)&#xFF0C;&#x5F97;&#x5230;&#x5982;&#x4E0B;&#x516C;&#x5F0F;(3)&#xFF1A;<br>
&#x540C;&#x6001;&#x6EE1;&#x8DB3;&#x4E0B;&#x9762;&#x6027;&#x8D28;&#xFF1A;</p>
<pre data-role="codeBlock" data-info class="language-"><code>f(a + b) = f(a) + f(b)
f(a + b) = f(a) * f(b)
</code></pre><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mstyle mathcolor="red"><mrow><mi mathvariant="normal">&#x4ECD;</mi><mi mathvariant="normal">&#x6709;</mi><mi mathvariant="normal">&#x7591;</mi><mi mathvariant="normal">&#x95EE;</mi></mrow></mstyle></mrow><annotation encoding="application/x-tex">\color{red}{&#x4ECD;&#x6709;&#x7591;&#x95EE;}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0em;vertical-align:0em;"></span><span class="mord" style="color:red;"><span class="mord cjk_fallback" style="color:red;">&#x4ECD;</span><span class="mord cjk_fallback" style="color:red;">&#x6709;</span><span class="mord cjk_fallback" style="color:red;">&#x7591;</span><span class="mord cjk_fallback" style="color:red;">&#x95EE;</span></span></span></span></span>&#xFF0C;&#x516C;&#x5F0F;(3)&#x600E;&#x4E48;&#x5F97;&#x51FA;&#x7684;<br>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mtable side="right"><mlabeledtr><mtd><mtext>(3)</mtext></mtd><mtd><mrow><mi>F</mi><mrow><mo fence="true">(</mo><msup><mrow><mo fence="true">(</mo><msub><mi>w</mi><mi>i</mi></msub><mo>&#x2212;</mo><msub><mi>w</mi><mi>j</mi></msub><mo fence="true">)</mo></mrow><mi>T</mi></msup><msub><mover accent="true"><mi>w</mi><mo>~</mo></mover><mi>k</mi></msub><mo fence="true">)</mo></mrow><mo>=</mo><mfrac><mrow><mi>F</mi><mrow><mo fence="true">(</mo><msubsup><mi>w</mi><mi>i</mi><mi>T</mi></msubsup><msub><mover accent="true"><mi>w</mi><mo>~</mo></mover><mi>k</mi></msub><mo fence="true">)</mo></mrow></mrow><mrow><mi>F</mi><mrow><mo fence="true">(</mo><msubsup><mi>w</mi><mi>j</mi><mi>T</mi></msubsup><msub><mover accent="true"><mi>w</mi><mo>~</mo></mover><mi>k</mi></msub><mo fence="true">)</mo></mrow></mrow></mfrac></mrow></mtd></mlabeledtr></mtable><annotation encoding="application/x-tex">F\left(\left(w_{i}-w_{j}\right)^{T} \tilde{w}_{k}\right)=\frac{F\left(w_{i}^{T} \tilde{w}_{k}\right)}{F\left(w_{j}^{T} \tilde{w}_{k}\right)} \tag{3}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.80002em;vertical-align:-0.65002em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">F</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size2">(</span></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">&#x2212;</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.981231em;"><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span></span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.16666em;">~</span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size2">)</span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.7429819999999996em;vertical-align:-1.1529719999999999em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.59001em;"><span style="top:-2.26em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">F</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.823131em;"><span style="top:-2.4231360000000004em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span><span style="top:-3.0448000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.4129719999999999em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span></span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.16666em;">~</span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.7400100000000003em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">F</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-2.441336em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span></span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.16666em;">~</span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:1.1529719999999999em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="tag"><span class="strut" style="height:2.7429819999999996em;vertical-align:-1.1529719999999999em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">3</span></span><span class="mord">)</span></span></span></span></span></span><br>
&#x6839;&#x636E;&#x516C;&#x5F0F;(2)&#x5F97;&#x4E0B;&#x9762;&#x516C;&#x5F0F;&#xFF1A;<br>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mtable side="right"><mlabeledtr><mtd><mtext>(4)</mtext></mtd><mtd><mrow><mi>F</mi><mrow><mo fence="true">(</mo><msubsup><mi>w</mi><mi>i</mi><mi>T</mi></msubsup><msub><mover accent="true"><mi>w</mi><mo>~</mo></mover><mi>k</mi></msub><mo fence="true">)</mo></mrow><mo>=</mo><msub><mi>P</mi><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo>=</mo><mfrac><msub><mi>X</mi><mrow><mi>i</mi><mi>k</mi></mrow></msub><msub><mi>X</mi><mi>i</mi></msub></mfrac></mrow></mtd></mlabeledtr></mtable><annotation encoding="application/x-tex">F\left(w_{i}^{T} \tilde{w}_{k}\right)=P_{i k}=\frac{X_{i k}}{X_{i}} \tag{4}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2413409999999998em;vertical-align:-0.35001em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">F</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span></span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.16666em;">~</span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.19633em;vertical-align:-0.8360000000000001em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.36033em;"><span style="top:-2.3139999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="tag"><span class="strut" style="height:2.19633em;vertical-align:-0.8360000000000001em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">4</span></span><span class="mord">)</span></span></span></span></span></span><br>
&#x5219;&#x516C;&#x5F0F;(3)&#x7684;&#x89E3;&#x4E3A;<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>F</mi><mo>=</mo><mi>e</mi><mi>x</mi><mi>p</mi><mo>(</mo><mo>)</mo></mrow><annotation encoding="application/x-tex">F=exp()</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">F</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">e</span><span class="mord mathdefault">x</span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mclose">)</span></span></span></span>&#xFF1A;<br>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mtable side="right"><mlabeledtr><mtd><mtext>(5)</mtext></mtd><mtd><mrow><msubsup><mi>w</mi><mi>i</mi><mi>T</mi></msubsup><msub><mover accent="true"><mi>w</mi><mo>~</mo></mover><mi>k</mi></msub><mo>=</mo><mi>log</mi><mo>&#x2061;</mo><mrow><mo fence="true">(</mo><msub><mi>P</mi><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo fence="true">)</mo></mrow><mo>=</mo><mi>log</mi><mo>&#x2061;</mo><mrow><mo fence="true">(</mo><msub><mi>X</mi><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo fence="true">)</mo></mrow><mo>&#x2212;</mo><mi>log</mi><mo>&#x2061;</mo><mrow><mo fence="true">(</mo><msub><mi>X</mi><mi>i</mi></msub><mo fence="true">)</mo></mrow></mrow></mtd></mlabeledtr></mtable><annotation encoding="application/x-tex">w_{i}^{T} \tilde{w}_{k}=\log \left(P_{i k}\right)=\log \left(X_{i k}\right)-\log \left(X_{i}\right) \tag{5}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.138331em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span></span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.16666em;">~</span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">&#x2212;</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span></span><span class="tag"><span class="strut" style="height:1.1413309999999999em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">5</span></span><span class="mord">)</span></span></span></span></span></span><br>
&#x4E0A;&#x8FF0;&#x516C;&#x5F0F;&#x4E2D;<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>l</mi><mi>o</mi><mi>g</mi><mo>(</mo><msub><mi>X</mi><mi>i</mi></msub><mo>)</mo></mrow><annotation encoding="application/x-tex">log(X_i)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.01968em;">l</span><span class="mord mathdefault">o</span><span class="mord mathdefault" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>&#x7684;&#x5B58;&#x5728;&#x7834;&#x574F;&#x4E86;&#x53EF;&#x4EA4;&#x6362;&#x5BF9;&#x79F0;&#x6027;(exchange symmetry)&#xFF0C;&#x4F46;&#x8BE5;&#x9879;&#x662F;&#x72EC;&#x7ACB;&#x4E8E;<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.03148em;">k</span></span></span></span>&#x7684;&#xFF0C;&#x6240;&#x4EE5;&#x4F7F;&#x7528;&#x4E0B;&#x9762;&#x7684;&#x516C;&#x5F0F;&#x6765;&#x5B9E;&#x73B0;&#x53EF;&#x4EA4;&#x6362;&#x5BF9;&#x79F0;&#xFF1A;<br>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mtable side="right"><mlabeledtr><mtd><mtext>(6)</mtext></mtd><mtd><mrow><msubsup><mi>w</mi><mi>i</mi><mi>T</mi></msubsup><msub><mover accent="true"><mi>w</mi><mo>~</mo></mover><mi>k</mi></msub><mo>+</mo><msub><mi>b</mi><mi>i</mi></msub><mo>+</mo><msub><mover accent="true"><mi>b</mi><mo>~</mo></mover><mi>k</mi></msub><mo>=</mo><mi>log</mi><mo>&#x2061;</mo><mrow><mo fence="true">(</mo><msub><mi>X</mi><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo fence="true">)</mo></mrow></mrow></mtd></mlabeledtr></mtable><annotation encoding="application/x-tex">w_{i}^{T} \tilde{w}_{k}+b_{i}+\tilde{b}_{k}=\log \left(X_{i k}\right) \tag{6}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.138331em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span></span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.16666em;">~</span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.0812999999999997em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9312999999999998em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">b</span></span></span><span style="top:-3.61344em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;">~</span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span></span><span class="tag"><span class="strut" style="height:1.1812999999999998em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">6</span></span><span class="mord">)</span></span></span></span></span></span><br>
&#x5176;&#x4E2D;<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>b</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">b_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>&#x4E0E;<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>l</mi><mi>o</mi><mi>g</mi><mo>(</mo><msub><mi>X</mi><mi>i</mi></msub><mo>)</mo></mrow><annotation encoding="application/x-tex">log(X_i)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.01968em;">l</span><span class="mord mathdefault">o</span><span class="mord mathdefault" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>&#x7B49;&#x4EF7;&#xFF0C;<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mover accent="true"><mi>b</mi><mo>~</mo></mover><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\tilde{b}_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0812999999999997em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9312999999999998em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">b</span></span></span><span style="top:-3.61344em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;">~</span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>&#x7684;&#x5B58;&#x5728;&#x662F;&#x4E3A;&#x4E86;&#x4FDD;&#x8BC1;<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mover accent="true"><mi>w</mi><mo>~</mo></mover></mrow><annotation encoding="application/x-tex">\tilde{w}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6678599999999999em;vertical-align:0em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span></span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.16666em;">~</span></span></span></span></span></span></span></span></span>&#x7684;&#x5BF9;&#x79F0;&#x6027;&#xFF0C;&#x56E0;&#x6B64;&#x4EA4;&#x6362;<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord mathdefault">i</span></span></span></span>&#x548C;<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.03148em;">k</span></span></span></span>&#x7684;&#x503C;&#xFF0C;&#x4E0A;&#x8FF0;&#x516C;&#x5F0F;&#x7684;&#x7ED3;&#x679C;&#x5B8C;&#x5168;&#x4E0D;&#x53D8;&#x3002;<br>
(3)&#x786E;&#x5B9A;&#x635F;&#x5931;&#x51FD;&#x6570;&#x5F62;&#x5F0F;<br>
&#x5728;&#x516C;&#x5F0F;(6)&#x4E2D;&#x5B58;&#x5728;&#x4E24;&#x4E2A;&#x95EE;&#x9898;&#xFF1A;</p>
<ol>
<li>&#x5F53;<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>X</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">X_{ij}=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span>&#x65F6;&#x51FD;&#x6570;&#x662F;&#x65E0;&#x610F;&#x4E49;&#x7684;&#xFF0C;&#x4F46;&#x5728;&#x5B9E;&#x9645;&#x4E2D;&#x5B58;&#x5728;&#x5927;&#x91CF;&#x7684;&#x8BCD;&#x76F4;&#x63A5;&#x662F;&#x6CA1;&#x6709;&#x5171;&#x73B0;&#x5173;&#x7CFB;&#x7684;&#xFF0C;&#x56E0;&#x6B64;&#x7B49;&#x4E8E;0&#x7684;&#x73B0;&#x8C61;&#x5927;&#x91CF;&#x5B58;&#x5728;&#xFF0C;&#x53EF;&#x4EE5;&#x8003;&#x8651;&#x7ED9;&#x5B83;&#x52A0;&#x4E2A;1&#xFF1A;<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>l</mi><mi>o</mi><mi>g</mi><mo>(</mo><msub><mi>X</mi><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo>)</mo><mo>&#x2192;</mo><mi>l</mi><mi>o</mi><mi>g</mi><mo>(</mo><mn>1</mn><mo>+</mo><msub><mi>X</mi><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo>)</mo></mrow><annotation encoding="application/x-tex">log(X_{ik}) \rightarrow log(1+X_{ik})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.01968em;">l</span><span class="mord mathdefault">o</span><span class="mord mathdefault" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&#x2192;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.01968em;">l</span><span class="mord mathdefault">o</span><span class="mord mathdefault" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></li>
<li>&#x8BE5;&#x516C;&#x5F0F;&#x4E2D;&#x6240;&#x6709;&#x7684;&#x5171;&#x73B0;&#x7684;&#x7ED3;&#x679C;&#x89C6;&#x4E3A;&#x76F8;&#x540C;&#x7684;(weights all co-occurrences equally)<br>
&#x5B9E;&#x9645;&#x4E2D;&#x6709;&#x7684;&#x5355;&#x8BCD;&#x95F4;&#x5171;&#x73B0;&#x5F88;&#x5C11;&#x751A;&#x81F3;&#x6CA1;&#x6709;&#xFF0C;&#x6240;&#x4EE5;&#x6BCF;&#x4E2A;&#x5171;&#x73B0;&#x90FD;&#x5E94;&#x8BE5;&#x4F7F;&#x7528;&#x4E0D;&#x540C;&#x7684;&#x6743;&#x91CD;</li>
</ol>
<p>&#x7EFC;&#x4E0A;&#x5F97;&#x5230;&#x4E0B;&#x9762;&#x516C;&#x5F0F;&#xFF1A;<br>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mtable side="right"><mlabeledtr><mtd><mtext>(7)</mtext></mtd><mtd><mrow><mi>J</mi><mo>=</mo><munderover><mo>&#x2211;</mo><mrow><mi>i</mi><mo separator="true">,</mo><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>V</mi></munderover><mi>f</mi><mrow><mo fence="true">(</mo><msub><mi>X</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo fence="true">)</mo></mrow><msup><mrow><mo fence="true">(</mo><msubsup><mi>w</mi><mi>i</mi><mi>T</mi></msubsup><msub><mover accent="true"><mi>w</mi><mo>~</mo></mover><mi>j</mi></msub><mo>+</mo><msub><mi>b</mi><mi>i</mi></msub><mo>+</mo><msub><mover accent="true"><mi>b</mi><mo>~</mo></mover><mi>j</mi></msub><mo>&#x2212;</mo><mi>log</mi><mo>&#x2061;</mo><msub><mi>X</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo fence="true">)</mo></mrow><mn>2</mn></msup></mrow></mtd></mlabeledtr></mtable><annotation encoding="application/x-tex">J=\sum_{i, j=1}^{V} f\left(X_{i j}\right)\left(w_{i}^{T} \tilde{w}_{j}+b_{i}+\tilde{b}_{j}-\log X_{i j}\right)^{2} \tag{7}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.09618em;">J</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:3.2421130000000007em;vertical-align:-1.4137769999999998em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000006em;"><span style="top:-1.872331em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">&#x2211;</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.22222em;">V</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:1.4137769999999998em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size2">(</span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span></span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.16666em;">~</span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9312999999999998em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">b</span></span></span><span style="top:-3.61344em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;">~</span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">&#x2212;</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size2">)</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.3540079999999999em;"><span style="top:-3.6029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span class="tag"><span class="strut" style="height:3.2421130000000007em;vertical-align:-1.4137769999999998em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">7</span></span><span class="mord">)</span></span></span></span></span></span><br>
&#x5176;&#x4E2D;<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.22222em;">V</span></span></span></span>&#x662F;&#x5B57;&#x5178;&#x7684;&#x5927;&#x5C0F;&#xFF0C;&#x4E0A;&#x8FF0;&#x516C;&#x5F0F;&#x4E2D;&#x635F;&#x5931;&#x51FD;&#x6570;&#x8D8B;&#x4E8E;&#x4F7F;&#x516C;&#x5F0F;(6)&#x7684;&#x7ED3;&#x679C;&#x6210;&#x7ACB;&#xFF0C;&#x6743;&#x91CD;&#x51FD;&#x6570;<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span></span></span></span>&#x6EE1;&#x8DB3;&#x4E0B;&#x9762;&#x7684;&#x5C5E;&#x6027;&#xFF1A;</p>
<ol>
<li><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>f</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f(0)=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span>&#xFF0C;&#x4FDD;&#x8BC1;&#x5728;&#x5355;&#x8BCD;&#x5171;&#x73B0;&#x8BA1;&#x6570;&#x4E3A;0&#x65F6;&#x4E0D;&#x4F1A;&#x4EA7;&#x751F;&#x635F;&#x5931;</li>
<li><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mclose">)</span></span></span></span>&#x5E94;&#x8BE5;&#x662F;&#x975E;&#x51CF;&#x51FD;&#x6570;&#xFF0C;&#x8FD9;&#x6837;&#x5171;&#x73B0;&#x7684;&#x4F4E;&#x9891;&#x8BCD;&#x7684;&#x6743;&#x91CD;&#x4E0D;&#x4F1A;&#x504F;&#x5927;(rare co-occurrences are not overweighted)</li>
<li><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mclose">)</span></span></span></span>&#x5BF9;&#x4E8E;&#x5927;&#x7684;&#x5171;&#x73B0;&#x8BA1;&#x6570;&#x5E94;&#x8BE5;&#x76F8;&#x5BF9;&#x8F83;&#x5C0F;&#xFF0C;&#x4E3A;&#x4E86;&#x9632;&#x6B62;&#x9AD8;&#x9891;&#x8BCD;&#x7684;&#x6743;&#x91CD;&#x8FC7;&#x5927;( frequent co-occurrences are not overweighted)</li>
</ol>
<p>&#x8FD9;&#x91CC;&#x7684;<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mclose">)</span></span></span></span>&#x7684;&#x5F62;&#x5F0F;&#x5982;&#x4E0B;&#xFF1A;<br>
<img src="imgs/function.jpg" alt><br>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mtable side="right"><mlabeledtr><mtd><mtext>(8)</mtext></mtd><mtd><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mrow><mo fence="true">{</mo><mtable><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msup><mrow><mo fence="true">(</mo><mi>x</mi><mi mathvariant="normal">/</mi><msub><mi>x</mi><mi>max</mi><mo>&#x2061;</mo></msub><mo fence="true">)</mo></mrow><mi>&#x3B1;</mi></msup></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>&#xA0;if&#xA0;</mtext><mi>x</mi><mo>&lt;</mo><msub><mi>x</mi><mi>max</mi><mo>&#x2061;</mo></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext>&#xA0;otherwise&#xA0;</mtext></mstyle></mtd></mtr></mtable></mrow></mrow></mtd></mlabeledtr></mtable><annotation encoding="application/x-tex">f(x)=\left\{\begin{array}{cl}{\left(x / x_{\max }\right)^{\alpha}} &amp; {\text { if } x&lt;x_{\max }} \\ {1} &amp; {\text { otherwise }}\end{array}\right. \tag{8}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.40003em;vertical-align:-0.95003em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">{</span></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord mathdefault">x</span><span class="mord">/</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.151392em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.301392em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mop mtight">max</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.804292em;"><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.0037em;">&#x3B1;</span></span></span></span></span></span></span></span></span></span></span></span><span style="top:-2.4099999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">1</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.9500000000000004em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord text"><span class="mord">&#xA0;if&#xA0;</span></span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.151392em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.301392em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mop mtight">max</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-2.4099999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord text"><span class="mord">&#xA0;otherwise&#xA0;</span></span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.9500000000000004em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="tag"><span class="strut" style="height:2.40003em;vertical-align:-0.95003em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">8</span></span><span class="mord">)</span></span></span></span></span></span><br>
&#x8FD9;&#x91CC;&#x7684;<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>&#x3B1;</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.0037em;">&#x3B1;</span></span></span></span>&#x53D6;<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>3</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">3/4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">3</span><span class="mord">/</span><span class="mord">4</span></span></span></span>&#xFF0C;&#x8FD9;&#x4E2A;&#x503C;&#x4E0E;&#x8D1F;&#x91C7;&#x6837;&#x8BBA;&#x6587;&#x4E2D;&#x7528;&#x4E00;&#x5143;&#x6A21;&#x578B;&#x7684;&#x6307;&#x6570;&#x4E00;&#x6837;&#xFF0C;<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>x</mi><mrow><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub><mo>=</mo><mn>100</mn></mrow><annotation encoding="application/x-tex">x_{max}=100</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">m</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">x</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord">0</span><span class="mord">0</span></span></span></span>&#x3002;</p>
<h3 class="mume-header" id="3-%E6%A8%A1%E5%9E%8B%E5%85%B3%E7%B3%BB">3. &#x6A21;&#x578B;&#x5173;&#x7CFB;</h3>

<p>&#x6240;&#x6709;&#x65E0;&#x76D1;&#x7763;&#x7684;&#x8BCD;&#x5411;&#x91CF;&#x5B66;&#x4E60;&#x65B9;&#x6CD5;&#x90FD;&#x662F;&#x57FA;&#x4E8E;&#x8BED;&#x6599;&#x7684;&#x5171;&#x73B0;&#x7279;&#x6027;&#x7684;&#xFF0C;&#x6240;&#x4EE5;&#x6A21;&#x578B;&#x95F4;&#x5E94;&#x8BE5;&#x6709;&#x4E00;&#x5B9A;&#x7684;&#x5171;&#x6027;<br>
&#x5728;skip-gram&#x6A21;&#x578B;&#x4E2D;&#xFF0C;&#x5355;&#x8BCD;<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.05724em;">j</span></span></span></span>&#x51FA;&#x73B0;&#x5728;&#x5355;&#x8BCD;<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord mathdefault">i</span></span></span></span>&#x7684;&#x4E0A;&#x4E0B;&#x6587;&#x4E2D;&#x7684;&#x9884;&#x6D4B;&#x6982;&#x7387;&#x4F7F;&#x7528;&#x4E0B;&#x9762;&#x5F0F;&#x5B50;&#xFF1A;<br>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mtable side="right"><mlabeledtr><mtd><mtext>(9)</mtext></mtd><mtd><mrow><msub><mi>Q</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mfrac><mrow><mi>exp</mi><mo>&#x2061;</mo><mrow><mo fence="true">(</mo><msubsup><mi>w</mi><mi>i</mi><mi>T</mi></msubsup><msub><mover accent="true"><mi>w</mi><mo>~</mo></mover><mi>j</mi></msub><mo fence="true">)</mo></mrow></mrow><mrow><munderover><mo>&#x2211;</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>V</mi></munderover><mi>exp</mi><mo>&#x2061;</mo><mrow><mo fence="true">(</mo><msubsup><mi>w</mi><mi>i</mi><mi>T</mi></msubsup><msub><mover accent="true"><mi>w</mi><mo>~</mo></mover><mi>k</mi></msub><mo fence="true">)</mo></mrow></mrow></mfrac></mrow></mtd></mlabeledtr></mtable><annotation encoding="application/x-tex">Q_{i j}=\frac{\exp \left(w_{i}^{T} \tilde{w}_{j}\right)}{\sum_{k=1}^{V} \exp \left(w_{i}^{T} \tilde{w}_{k}\right)} \tag{9}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.811251em;vertical-align:-1.221241em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.59001em;"><span style="top:-2.128769em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em;">&#x2211;</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.981231em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.22222em;">V</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.29971000000000003em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">exp</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8231310000000001em;"><span style="top:-2.4231360000000004em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.0448000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.27686399999999994em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span></span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.16666em;">~</span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.7400100000000003em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop">exp</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-2.441336em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span></span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.16666em;">~</span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:1.221241em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="tag"><span class="strut" style="height:2.811251em;vertical-align:-1.221241em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">9</span></span><span class="mord">)</span></span></span></span></span></span><br>
&#x4F7F;&#x7528;&#x8BCD;&#x5411;&#x91CF;&#x70B9;&#x79EF;&#x7136;&#x540E;&#x5F52;&#x4E00;&#x5316;&#x7684;&#x65B9;&#x5F0F;&#x6765;&#x8BA1;&#x7B97;&#x4E0A;&#x4E0B;&#x6587;&#x4E2D;&#x67D0;&#x4E2A;&#x8BCD;&#x7684;&#x6982;&#x7387;&#xFF0C;&#x6240;&#x4EE5;&#x76EE;&#x6807;&#x51FD;&#x6570;&#x5C31;&#x662F;&#x6700;&#x5927;&#x5316;&#x6240;&#x6709;&#x7684;&#x4E0A;&#x4E0B;&#x6587;&#x6982;&#x7387;&#xFF0C;&#x65B9;&#x6CD5;&#x5C31;&#x662F;&#x5C06;&#x6240;&#x6709;&#x8BCD;&#x7684;&#x4E0A;&#x4E0B;&#x6587;&#x6982;&#x7387;&#x76F8;&#x4E58;&#xFF0C;&#x7136;&#x540E;&#x6C42;&#x6700;&#x5927;&#x503C;&#xFF0C;&#x4E00;&#x822C;&#x53D6;&#x8D1F;&#x5BF9;&#x6570;&#x53D8;&#x4E3A;&#x6C42;&#x548C;&#xFF0C;&#x516C;&#x5F0F;&#x5982;&#x4E0B;&#xFF1A;<br>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mtable side="right"><mlabeledtr><mtd><mtext>(10)</mtext></mtd><mtd><mrow><mi>J</mi><mo>=</mo><mo>&#x2212;</mo><munder><mo>&#x2211;</mo><mfrac linethickness="0px"><mrow><mi>i</mi><mo>&#x2208;</mo><mtext>&#xA0;corpus&#xA0;</mtext></mrow><mrow><mi>j</mi><mo>&#x2208;</mo><mi mathvariant="normal">context</mi><mo>&#x2061;</mo><mo>(</mo><mi>i</mi><mo>)</mo></mrow></mfrac></munder><mi>log</mi><mo>&#x2061;</mo><msub><mi>Q</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></mtd></mlabeledtr></mtable><annotation encoding="application/x-tex">J=-\sum_{i \in \text { corpus } \atop j \in \operatorname{context}(i)} \log Q_{i j} \tag{10}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.09618em;">J</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.89589em;vertical-align:-1.845885em;"></span><span class="mord">&#x2212;</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.6722249999999999em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9453857142857143em;"><span style="top:-2.1884142857142854em;"><span class="pstrut" style="height:2.5357142857142856em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span><span class="mrel mtight">&#x2208;</span><span class="mop mtight"><span class="mord mathrm mtight">c</span><span class="mord mathrm mtight">o</span><span class="mord mathrm mtight">n</span><span class="mord mathrm mtight">t</span><span class="mord mathrm mtight">e</span><span class="mord mathrm mtight">x</span><span class="mord mathrm mtight">t</span></span><span class="mopen mtight">(</span><span class="mord mathdefault mtight">i</span><span class="mclose mtight">)</span></span></span></span><span style="top:-3.0100142857142855em;"><span class="pstrut" style="height:2.5357142857142856em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mrel mtight">&#x2208;</span><span class="mord text mtight"><span class="mord mtight">&#xA0;corpus&#xA0;</span></span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.5258714285714285em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">&#x2211;</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:1.845885em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span><span class="tag"><span class="strut" style="height:2.89589em;vertical-align:-1.845885em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">1</span><span class="mord">0</span></span><span class="mord">)</span></span></span></span></span></span><br>
&#x4E0A;&#x9762;&#x516C;&#x5F0F;&#x4E2D;&#x5F52;&#x4E00;&#x5316;&#x5206;&#x6BCD;&#x8BA1;&#x7B97;&#x91CF;&#x5F88;&#x5927;&#xFF0C;<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mstyle mathcolor="red"><mrow><mi>s</mi><mi>k</mi><mi>i</mi><mi>p</mi><mo>&#x2212;</mo><mi>g</mi><mi>r</mi><mi>a</mi><mi>m</mi><mi mathvariant="normal">&#x4E2D;</mi><mi mathvariant="normal">&#x4F1A;</mi><mi mathvariant="normal">&#x5F15;</mi><mi mathvariant="normal">&#x5165;</mi><mi mathvariant="normal">&#x8FD1;</mi><mi mathvariant="normal">&#x4F3C;</mi><mi mathvariant="normal">&#x7684;</mi><mi mathvariant="normal">&#x503C;</mi></mrow></mstyle></mrow><annotation encoding="application/x-tex">\color{red}{skip-gram&#x4E2D;&#x4F1A;&#x5F15;&#x5165;&#x8FD1;&#x4F3C;&#x7684;&#x503C;}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord" style="color:red;"><span class="mord mathdefault" style="color:red;">s</span><span class="mord mathdefault" style="margin-right:0.03148em;color:red;">k</span><span class="mord mathdefault" style="color:red;">i</span><span class="mord mathdefault" style="color:red;">p</span><span class="mspace" style="color:red;margin-right:0.2222222222222222em;"></span><span class="mbin" style="color:red;">&#x2212;</span><span class="mspace" style="color:red;margin-right:0.2222222222222222em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;color:red;">g</span><span class="mord mathdefault" style="margin-right:0.02778em;color:red;">r</span><span class="mord mathdefault" style="color:red;">a</span><span class="mord mathdefault" style="color:red;">m</span><span class="mord cjk_fallback" style="color:red;">&#x4E2D;</span><span class="mord cjk_fallback" style="color:red;">&#x4F1A;</span><span class="mord cjk_fallback" style="color:red;">&#x5F15;</span><span class="mord cjk_fallback" style="color:red;">&#x5165;</span><span class="mord cjk_fallback" style="color:red;">&#x8FD1;</span><span class="mord cjk_fallback" style="color:red;">&#x4F3C;</span><span class="mord cjk_fallback" style="color:red;">&#x7684;</span><span class="mord cjk_fallback" style="color:red;">&#x503C;</span></span></span></span></span>&#xFF0C;&#x8BE5;&#x6A21;&#x578B;&#x9700;&#x8981;&#x6309;&#x987A;&#x5E8F;&#x904D;&#x5386;&#x8BED;&#x6599;&#xFF0C;&#x6240;&#x4EE5;&#x8BAD;&#x7EC3;&#x7684;&#x6548;&#x7387;&#x5F88;&#x4F4E;&#xFF0C;&#x53EF;&#x4EE5;&#x76F4;&#x63A5;&#x7EDF;&#x8BA1;&#x6240;&#x6709;&#x7684;&#x5355;&#x8BCD;&#x5171;&#x73B0;&#x6B21;&#x6570;&#xFF0C;&#x5C06;&#x516C;&#x5F0F;(10)&#x8F6C;&#x6362;&#x4E3A;&#x4E0B;&#x9762;&#x7684;&#x5F62;&#x5F0F;&#xFF1A;<br>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mtable side="right"><mlabeledtr><mtd><mtext>(11)</mtext></mtd><mtd><mrow><mi>J</mi><mo>=</mo><mo>&#x2212;</mo><munderover><mo>&#x2211;</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>V</mi></munderover><munderover><mo>&#x2211;</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>V</mi></munderover><msub><mi>X</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mi>log</mi><mo>&#x2061;</mo><msub><mi>Q</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></mtd></mlabeledtr></mtable><annotation encoding="application/x-tex">J=-\sum_{i=1}^{V} \sum_{j=1}^{V} X_{i j} \log Q_{i j} \tag{11}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.09618em;">J</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:3.2421130000000007em;vertical-align:-1.4137769999999998em;"></span><span class="mord">&#x2212;</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000002em;"><span style="top:-1.872331em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">&#x2211;</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.22222em;">V</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:1.277669em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000006em;"><span style="top:-1.872331em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">&#x2211;</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.22222em;">V</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:1.4137769999999998em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span><span class="tag"><span class="strut" style="height:3.2421130000000007em;vertical-align:-1.4137769999999998em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">1</span><span class="mord">1</span></span><span class="mord">)</span></span></span></span></span></span><br>
&#x5F53;&#x4E24;&#x4E2A;&#x5355;&#x8BCD;&#x6CA1;&#x6709;&#x51FA;&#x73B0;&#x5171;&#x73B0;&#x65F6;<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>X</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">X_{ij}=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span>&#xFF0C;&#x6240;&#x4EE5;&#x4E0D;&#x4F1A;&#x4EA7;&#x751F;&#x635F;&#x5931;&#xFF0C;&#x53EA;&#x5BF9;&#x5B58;&#x5728;&#x4E0A;&#x4E0B;&#x6587;&#x5171;&#x73B0;&#x5173;&#x7CFB;&#x7684;&#x8BCD;&#x8BA1;&#x7B97;&#x635F;&#x5931;&#xFF0C;&#x8FD9;&#x6837;&#x548C;&#x516C;&#x5F0F;(10)&#x5B9E;&#x73B0;&#x7684;&#x4F5C;&#x7528;&#x662F;&#x76F8;&#x540C;&#x7684;&#x3002;<br>
&#x6839;&#x636E;<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>X</mi><mi>i</mi></msub><mo>=</mo><msub><mo>&#x2211;</mo><mi>k</mi></msub><msub><mi>X</mi><mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow><annotation encoding="application/x-tex">X_{i}=\sum_{k} X_{i k}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.0497100000000001em;vertical-align:-0.29971000000000003em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em;">&#x2211;</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1863979999999999em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.29971000000000003em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>&#x548C;<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>P</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><msub><mi>X</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mi mathvariant="normal">/</mi><msub><mi>X</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">P_{i j}=X_{i j} / X_{i}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.036108em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mord">/</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>&#x5C06;&#x516C;&#x5F0F;(11)&#x8F6C;&#x6362;&#x4E3A;&#x4E0B;&#x9762;&#x7684;&#x5F62;&#x5F0F;&#xFF1A;<br>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mtable side="right"><mlabeledtr><mtd><mtext>(12)</mtext></mtd><mtd><mrow><mi>J</mi><mo>=</mo><mo>&#x2212;</mo><munderover><mo>&#x2211;</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>V</mi></munderover><msub><mi>X</mi><mi>i</mi></msub><munderover><mo>&#x2211;</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>V</mi></munderover><msub><mi>P</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mi>log</mi><mo>&#x2061;</mo><msub><mi>Q</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><munderover><mo>&#x2211;</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>V</mi></munderover><msub><mi>X</mi><mi>i</mi></msub><mi>H</mi><mrow><mo fence="true">(</mo><msub><mi>P</mi><mi>i</mi></msub><mo separator="true">,</mo><msub><mi>Q</mi><mi>i</mi></msub><mo fence="true">)</mo></mrow></mrow></mtd></mlabeledtr></mtable><annotation encoding="application/x-tex">J=-\sum_{i=1}^{V} X_{i} \sum_{j=1}^{V} P_{i j} \log Q_{i j}=\sum_{i=1}^{V} X_{i} H\left(P_{i}, Q_{i}\right) \tag{12}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.09618em;">J</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:3.2421130000000007em;vertical-align:-1.4137769999999998em;"></span><span class="mord">&#x2212;</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000002em;"><span style="top:-1.872331em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">&#x2211;</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.22222em;">V</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:1.277669em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000006em;"><span style="top:-1.872331em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">&#x2211;</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.22222em;">V</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:1.4137769999999998em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:3.106005em;vertical-align:-1.277669em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000002em;"><span style="top:-1.872331em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">&#x2211;</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.22222em;">V</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:1.277669em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathdefault" style="margin-right:0.08125em;">H</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span></span><span class="tag"><span class="strut" style="height:3.2421130000000007em;vertical-align:-1.4137769999999998em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">1</span><span class="mord">2</span></span><span class="mord">)</span></span></span></span></span></span><br>
&#x5176;&#x4E2D;<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>H</mi><mrow><mo fence="true">(</mo><msub><mi>P</mi><mi>i</mi></msub><mo separator="true">,</mo><msub><mi>Q</mi><mi>i</mi></msub><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">H\left(P_{i}, Q_{i}\right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.08125em;">H</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span></span></span></span>&#x662F;<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>P</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">P_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>&#x548C;<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>Q</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">Q_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>&#x7684;&#x4EA4;&#x53C9;&#x71B5;&#xFF0C;<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>P</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">P_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>&#x662F;&#x901A;&#x8FC7;&#x5171;&#x73B0;&#x77E9;&#x9635;&#x8BA1;&#x7B97;&#x5F97;&#x5230;&#x7684;&#xFF0C;&#x53EF;&#x4EE5;&#x5F53;&#x4F5C;&#x771F;&#x5B9E;&#x7684;&#x5206;&#x5E03;&#xFF0C;<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>Q</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">Q_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>&#x662F;&#x901A;&#x8FC7;&#x8BCD;&#x5411;&#x91CF;&#x8BA1;&#x7B97;&#x7684;&#x662F;&#x9884;&#x6D4B;&#x5206;&#x5E03;&#x3002;<br>
&#x516C;&#x5F0F;(12)&#x4E2D;&#x8FD8;&#x5B58;&#x5728;&#x4E24;&#x4E2A;&#x95EE;&#x9898;&#xFF1A;</p>
<ol>
<li>&#x6A21;&#x578B;&#x4E3A;&#x6240;&#x6709;&#x5355;&#x8BCD;&#x90FD;&#x5206;&#x914D;&#x76F8;&#x540C;&#x7684;&#x6743;&#x91CD;&#xFF0C;&#x4E00;&#x4E9B;&#x4E0D;&#x53EF;&#x80FD;&#x4E8B;&#x4EF6;(unlikely events)&#xFF0C;&#x5373;&#x6CA1;&#x6709;&#x5171;&#x73B0;&#x7684;&#x5355;&#x8BCD;&#x4E5F;&#x53C2;&#x4E0E;&#x4E86;&#x8BA1;&#x7B97;</li>
<li><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">Q</span></span></span></span>&#x8FD8;&#x9700;&#x8981;&#x5F52;&#x4E00;&#x5316;&#x624D;&#x884C;&#xFF0C;&#x8BA1;&#x7B97;&#x91CF;&#x5F88;&#x5927;</li>
</ol>
<p>&#x89E3;&#x51B3;&#x65B9;&#x6CD5;&#xFF1A;<strong>&#x6BCF;&#x4E2A;&#x5206;&#x5E03;&#x52A0;&#x4E2A;&#x6743;&#x91CD;</strong> + <strong>&#x6700;&#x5C0F;&#x4E8C;&#x4E58;&#x76EE;&#x6807;&#x51FD;&#x6570;</strong>(&#x4E0D;&#x9700;&#x8981;&#x5F52;&#x4E00;&#x5316;)<br>
&#x7ED3;&#x679C;&#x5982;&#x4E0B;&#xFF1A;<br>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mtable side="right"><mlabeledtr><mtd><mtext>(13)</mtext></mtd><mtd><mrow><mover accent="true"><mi>J</mi><mo>^</mo></mover><mo>=</mo><munder><mo>&#x2211;</mo><mrow><mi>i</mi><mo separator="true">,</mo><mi>j</mi></mrow></munder><msub><mi>X</mi><mi>i</mi></msub><msup><mrow><mo fence="true">(</mo><msub><mover accent="true"><mi>P</mi><mo>^</mo></mover><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>&#x2212;</mo><msub><mover accent="true"><mi>Q</mi><mo>^</mo></mover><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo fence="true">)</mo></mrow><mn>2</mn></msup></mrow></mtd></mlabeledtr></mtable><annotation encoding="application/x-tex">\hat{J}=\sum_{i, j} X_{i}\left(\hat{P}_{i j}-\hat{Q}_{i j}\right)^{2} \tag{13}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9467699999999999em;vertical-align:0em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9467699999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.09618em;">J</span></span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.08332999999999999em;">^</span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.767785em;vertical-align:-1.413777em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.8723309999999997em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">&#x2211;</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:1.413777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size2">(</span></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9467699999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">P</span></span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.16666em;">^</span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">&#x2212;</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9467699999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">Q</span></span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.16666em;">^</span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em;"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size2">)</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.3540079999999999em;"><span style="top:-3.6029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span class="tag"><span class="strut" style="height:2.767785em;vertical-align:-1.413777em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">1</span><span class="mord">3</span></span><span class="mord">)</span></span></span></span></span></span><br>
&#x5176;&#x4E2D;<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mover accent="true"><mi>P</mi><mo>^</mo></mover><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><msub><mi>X</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\hat{P}_{i j}=X_{i j}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.232878em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9467699999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">P</span></span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.16666em;">^</span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span>&#xFF0C;<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mover accent="true"><mi>Q</mi><mo>^</mo></mover><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mi>exp</mi><mo>&#x2061;</mo><mrow><mo fence="true">(</mo><msubsup><mi>w</mi><mi>i</mi><mi>T</mi></msubsup><msub><mover accent="true"><mi>w</mi><mo>~</mo></mover><mi>j</mi></msub><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\hat{Q}_{i j}=\exp \left(w_{i}^{T} \tilde{w}_{j}\right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.232878em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9467699999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">Q</span></span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.16666em;">^</span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em;"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.20001em;vertical-align:-0.35001em;"></span><span class="mop">exp</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-2.441336em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span></span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.16666em;">~</span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span></span></span></span>&#xFF0C;&#x8FD9;&#x4E24;&#x4E2A;&#x90FD;&#x662F;&#x672A;&#x5F52;&#x4E00;&#x5316;&#x7684;&#x5206;&#x5E03;&#xFF0C;&#x4F46;&#x662F;<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>X</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">X_{ij}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span>&#x503C;&#x5F88;&#x5927;&#xFF0C;&#x6240;&#x4EE5;&#x9700;&#x8981;&#x52A0;&#x4E2A;&#x5BF9;&#x6570;&#x64CD;&#x4F5C;&#x6765;&#x964D;&#x4F4E;&#x503C;&#x7684;&#x91CF;&#x7EA7;&#xFF1A;<br>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mtable side="right"><mlabeledtr><mtd><mtext>(14)</mtext></mtd><mtd><mtable><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mover accent="true"><mi>J</mi><mo>^</mo></mover></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><munder><mo>&#x2211;</mo><mrow><mi>i</mi><mo separator="true">,</mo><mi>j</mi></mrow></munder><msub><mi>X</mi><mi>i</mi></msub><msup><mrow><mo fence="true">(</mo><mi>log</mi><mo>&#x2061;</mo><msub><mover accent="true"><mi>P</mi><mo>^</mo></mover><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>&#x2212;</mo><mi>log</mi><mo>&#x2061;</mo><msub><mover accent="true"><mi>Q</mi><mo>^</mo></mover><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo fence="true">)</mo></mrow><mn>2</mn></msup></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><munder><mo>&#x2211;</mo><mrow><mi>i</mi><mo separator="true">,</mo><mi>j</mi></mrow></munder><msub><mi>X</mi><mi>i</mi></msub><msup><mrow><mo fence="true">(</mo><msubsup><mi>w</mi><mi>i</mi><mi>T</mi></msubsup><msub><mover accent="true"><mi>w</mi><mo>~</mo></mover><mi>j</mi></msub><mo>&#x2212;</mo><mi>log</mi><mo>&#x2061;</mo><msub><mi>X</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo fence="true">)</mo></mrow><mn>2</mn></msup></mrow></mstyle></mtd></mtr></mtable></mtd></mlabeledtr></mtable><annotation encoding="application/x-tex">\begin{aligned} \hat{J} &amp;=\sum_{i, j} X_{i}\left(\log \hat{P}_{i j}-\log \hat{Q}_{i j}\right)^{2} \\ &amp;=\sum_{i, j} X_{i}\left(w_{i}^{T} \tilde{w}_{j}-\log X_{i j}\right)^{2} \end{aligned} \tag{14}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:5.876901em;vertical-align:-2.6884505em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.1884505em;"><span style="top:-5.1884505em;"><span class="pstrut" style="height:3.354008em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9467699999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.09618em;">J</span></span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.08332999999999999em;">^</span></span></span></span></span></span></span></span><span style="top:-2.3793344999999997em;"><span class="pstrut" style="height:3.354008em;"></span><span class="mord"></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:2.6884505em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.1884505em;"><span style="top:-5.1884505em;"><span class="pstrut" style="height:3.354008em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.8723309999999997em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">&#x2211;</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:1.413777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size2">(</span></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9467699999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">P</span></span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.16666em;">^</span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">&#x2212;</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9467699999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">Q</span></span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.16666em;">^</span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em;"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size2">)</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.3540079999999999em;"><span style="top:-3.6029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:-2.3793344999999997em;"><span class="pstrut" style="height:3.354008em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.8723309999999997em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">&#x2211;</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:1.413777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span></span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.16666em;">~</span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">&#x2212;</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0953389999999998em;"><span style="top:-3.344231em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:2.6884505em;"><span></span></span></span></span></span></span></span></span><span class="tag"><span class="strut" style="height:5.876901em;vertical-align:-2.6884505em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">1</span><span class="mord">4</span></span><span class="mord">)</span></span></span></span></span></span><br>
&#x5728;&#x4E0A;&#x9762;&#x7684;&#x516C;&#x5F0F;&#x4E2D;&#x6743;&#x91CD;<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>X</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">X_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>&#x662F;&#x904D;&#x5386;&#x8BED;&#x6599;&#x8BA1;&#x7B97;&#x5171;&#x73B0;&#x6B21;&#x6570;&#x5F97;&#x5230;&#x7684;&#xFF0C;&#x56E0;&#x6B64;&#x9AD8;&#x9891;&#x8BCD;&#x7684;&#x5171;&#x73B0;&#x8BA1;&#x6570;&#x4F1A;&#x5F88;&#x5927;&#xFF0C;&#x4F4E;&#x9891;&#x8BCD;&#x7684;&#x4F1A;&#x5F88;&#x5C0F;&#xFF0C;&#x5728;&#x8D1F;&#x91C7;&#x6837;&#x8BBA;&#x6587;&#x4E2D;&#x5BF9;&#x9AD8;&#x9891;&#x8BCD;&#x8FDB;&#x884C;&#x5B50;&#x91C7;&#x6837;(subsampling)&#x6765;&#x51CF;&#x5C11;&#x9AD8;&#x9891;&#x8BCD;&#x7684;&#x4F7F;&#x7528;&#x9891;&#x7387;&#xFF0C;&#x53EF;&#x4EE5;&#x63D0;&#x9AD8;&#x6027;&#x80FD;&#xFF0C;&#x56E0;&#x6B64;&#x8FD9;&#x91CC;&#x5C06;&#x6743;&#x91CD;<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>X</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">X_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>&#x53C2;&#x6570;&#x5316;&#x4F1A;&#x66F4;&#x5408;&#x7406;&#xFF0C;&#x516C;&#x5F0F;&#x5982;&#x4E0B;&#xFF1A;<br>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mtable side="right"><mlabeledtr><mtd><mtext>(15)</mtext></mtd><mtd><mrow><mover accent="true"><mi>J</mi><mo>^</mo></mover><mo>=</mo><munder><mo>&#x2211;</mo><mrow><mi>i</mi><mo separator="true">,</mo><mi>j</mi></mrow></munder><mi>f</mi><mrow><mo fence="true">(</mo><msub><mi>X</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo fence="true">)</mo></mrow><msup><mrow><mo fence="true">(</mo><msubsup><mi>w</mi><mi>i</mi><mi>T</mi></msubsup><msub><mover accent="true"><mi>w</mi><mo>~</mo></mover><mi>j</mi></msub><mo>&#x2212;</mo><mi>log</mi><mo>&#x2061;</mo><msub><mi>X</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo fence="true">)</mo></mrow><mn>2</mn></msup></mrow></mtd></mlabeledtr></mtable><annotation encoding="application/x-tex">\hat{J}=\sum_{i, j} f\left(X_{i j}\right)\left(w_{i}^{T} \tilde{w}_{j}-\log X_{i j}\right)^{2} \tag{15}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9467699999999999em;vertical-align:0em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9467699999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.09618em;">J</span></span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.08332999999999999em;">^</span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.5091159999999997em;vertical-align:-1.413777em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.8723309999999997em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">&#x2211;</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:1.413777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span></span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.16666em;">~</span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">&#x2212;</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0953389999999998em;"><span style="top:-3.344231em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span class="tag"><span class="strut" style="height:2.5091159999999997em;vertical-align:-1.413777em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">1</span><span class="mord">5</span></span><span class="mord">)</span></span></span></span></span></span><br>
&#x8FD9;&#x91CC;&#x5F97;&#x5230;&#x7684;&#x516C;&#x5F0F;(15)&#x548C;&#x524D;&#x9762;&#x63A8;&#x7406;&#x5F97;&#x5230;&#x7684;&#x516C;&#x5F0F;(7)&#x662F;&#x7B49;&#x4EF7;&#x7684;&#xFF0C;&#x5F53;&#x7136;&#x504F;&#x7F6E;&#x4E5F;&#x53EF;&#x4EE5;&#x5305;&#x542B;&#x5230;&#x516C;&#x5F0F;(15)&#x4E2D;&#x3002;</p>
<p>&#x603B;&#x7ED3;&#xFF1A;</p>
<ol>
<li>GloVe&#x4F7F;&#x7528;&#x5171;&#x73B0;&#x77E9;&#x9635;&#x8BA1;&#x7B97;&#x5171;&#x73B0;&#x6982;&#x7387;&#x7684;&#x6BD4;&#x7387;&#xFF0C;&#x901A;&#x8FC7;&#x4F18;&#x5316;&#x8FD9;&#x4E2A;&#x6BD4;&#x7387;&#x6765;&#x5F97;&#x5230;&#x8BCD;&#x5411;&#x91CF;&#xFF0C;&#x4F7F;&#x7528;&#x5171;&#x73B0;&#x77E9;&#x9635;&#x53EF;&#x4EE5;&#x5229;&#x7528;&#x5168;&#x5C40;&#x7684;&#x5171;&#x73B0;&#x7EDF;&#x8BA1;&#x4FE1;&#x606F;&#xFF0C;&#x8FD9;&#x91CC;&#x4F7F;&#x7528;&#x4E86;&#x57FA;&#x4E8E;&#x77E9;&#x9635;&#x5206;&#x89E3;&#x7684;&#x65B9;&#x6CD5;(LSA)&#x7684;&#x4F18;&#x70B9;&#xFF1B;<strong>&#x4F18;&#x5316;&#x5171;&#x73B0;&#x6982;&#x7387;&#x6BD4;&#x7387;</strong>&#x76F8;&#x5F53;&#x4E8E;&#x4F18;&#x5316;&#x5C40;&#x90E8;&#x4E0A;&#x4E0B;&#x6587;&#x7684;&#x6982;&#x7387;&#xFF0C;&#x8FD9;&#x91CC;&#x4F7F;&#x7528;&#x4E86;&#x57FA;&#x4E8E;&#x5C40;&#x90E8;&#x7A97;&#x53E3;&#x7684;&#x65B9;&#x6CD5;(skip-gram)&#x4E2D;&#x7684;&#x4F18;&#x70B9;&#xFF0C;&#x6240;&#x4EE5;GloVe&#x6548;&#x679C;&#x66F4;&#x597D;&#x3002;</li>
<li>&#x4F7F;&#x7528;&#x6700;&#x5C0F;&#x4E8C;&#x4E58;&#x7684;&#x76EE;&#x6807;&#x51FD;&#x6570;&#x6765;&#x8861;&#x91CF;&#x4E24;&#x4E2A;&#x5206;&#x5E03;&#x7684;&#x8DDD;&#x79BB;&#xFF0C;&#x7701;&#x53BB;&#x4E86;&#x4EA4;&#x53C9;&#x71B5;&#x4E2D;&#x5F52;&#x4E00;&#x5316;&#x7684;&#x5DE8;&#x5927;&#x7684;&#x8BA1;&#x7B97;&#x91CF;</li>
<li>&#x4E3A;&#x5171;&#x73B0;&#x5355;&#x8BCD;&#x5B89;&#x6392;&#x4E0D;&#x540C;&#x7684;&#x6743;&#x91CD;&#xFF0C;&#x867D;&#x7136;&#x8FD9;&#x4E2A;&#x6743;&#x91CD;&#x6CA1;&#x6709;&#x5F52;&#x4E00;&#x5316;&#xFF0C;&#x4F46;&#x503C;&#x5C5E;&#x4E8E;[0,1]&#xFF0C;&#x6743;&#x91CD;&#x8868;&#x793A;&#x4E0A;&#x4E0B;&#x6587;&#x4E2D;&#x6BCF;&#x4E2A;&#x5355;&#x8BCD;&#x5BF9;&#x4E2D;&#x5FC3;&#x8BCD;&#x7684;&#x5F71;&#x54CD;&#x662F;&#x4E0D;&#x540C;&#x7684;&#xFF0C;&#x6709;&#x70B9;attention&#x7684;&#x611F;&#x89C9;&#xFF0C;&#x9700;&#x8981;&#x6CE8;&#x610F;&#x7684;&#x662F;&#x5F53;&#x4E24;&#x4E2A;&#x5355;&#x8BCD;&#x4E0D;&#x5171;&#x73B0;&#x65F6;&#x6743;&#x91CD;&#x4E3A;0&#xFF0C;&#x5373;&#x4E0D;&#x8BA1;&#x5165;&#x6700;&#x7EC8;&#x7684;&#x635F;&#x5931;&#x3002;</li>
<li>&#x5728;&#x5171;&#x73B0;&#x77E9;&#x9635;<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.07847em;">X</span></span></span></span>&#x4E2D;&#xFF0C;&#x6BCF;&#x4E2A;&#x5355;&#x8BCD;&#x5C5E;&#x4E8E;&#x4E2D;&#x5FC3;&#x8BCD;&#x4E5F;&#x5C5E;&#x4E8E;&#x4E0A;&#x4E0B;&#x6587;&#xFF0C;&#x6240;&#x4EE5;&#x8BE5;&#x77E9;&#x9635;&#x662F;&#x5BF9;&#x79F0;&#x7684;&#xFF0C;&#x6A21;&#x578B;&#x9700;&#x8981;&#x6EE1;&#x8DB3;<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mstyle mathcolor="red"><mrow><mi mathvariant="normal">&#x540C;</mi><mi mathvariant="normal">&#x6001;</mi><mi mathvariant="normal">&#x6027;</mi></mrow></mstyle></mrow><annotation encoding="application/x-tex">\color{red}{&#x540C;&#x6001;&#x6027;}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0em;vertical-align:0em;"></span><span class="mord" style="color:red;"><span class="mord cjk_fallback" style="color:red;">&#x540C;</span><span class="mord cjk_fallback" style="color:red;">&#x6001;</span><span class="mord cjk_fallback" style="color:red;">&#x6027;</span></span></span></span></span>&#xFF0C;&#x4EA4;&#x6362;&#x4E24;&#x4E2A;&#x53D8;&#x91CF;&#x5BF9;&#x7ED3;&#x679C;&#x4E0D;&#x4F1A;&#x4EA7;&#x751F;&#x5F71;&#x54CD;&#x3002;</li>
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